3SUM Is Really Hard: A Real-to-Integer Reduction

Nick Fischer; Adam Polak; Jonas Schmidt

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arXiv · 2610.10153

3SUM Is Really Hard: A Real-to-Integer Reduction

Nick Fischer·Adam Polak·Jonas Schmidt

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Abstract

We show that if the 3SUM problem on integer-valued inputs can be solved in truly subquadratic time, then it can also be solved in truly subquadratic time on real-valued inputs. This answers an open problem posed by Chan, Vassilevska Williams, and Xu [STOC 2022], and constitutes the first such tight real-to-integer self-reduction in fine-grained complexity. Our proof relies on a surprising combination of the Frank--Tardos theorem from optimization with Freiman-type theorems from additive combinatorics.

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Nick Fischer, Adam Polak, Jonas Schmidt. 2026-10-07. 3SUM Is Really Hard: A Real-to-Integer Reduction. https://arxiv.org/abs/2610.10153

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